Author: Eiko

Time: 2025-03-04 13:04:45 - 2025-03-04 13:04:45 (UTC)

Let \(C/\mathbb{Q}\) be a curve of genus \(g\) over \(\mathbb{Q}\),

\[y^2 = x^{2g+1}+\dots +b_0\]

Adn let \(J=J(C)\) the Jacobian, we have \(J\subset \mathbb{P}^{3^g-1}\) via \(|3\cdot \theta|\).

• \(g=2\), \(J\subset \mathbb{P}^8\), birationally it is basically a symetric square of \(C\), \(J\sim \mathrm{Sym}^2 C\)., $PJ() $ is represented by \(\{P_1,P_2\}, P_i\in C(\mathbb{Q})\), actually \(P_1+P_2-2\infty\).

\[\theta = \{P, -\infty, P\}\]

\(V_9=\mathcal{O}(9)\) is the space of linear forms on my projective space \(\mathbb{P}^8\), consider

\[\wedge^3 V_9 \to \wedge V_9 \otimes V_9\]

for example \(e_1\wedge e_2\wedge e_3\mapsto x_1\otimes e_2\wedge e_3 - x_2 \otimes e_1\wedge e_2 + x_3 \otimes e_2\wedge e_3\).

\[\begin{pmatrix} 0 & x_1 & -x_3 \\ -x_1 & 0 & x_2 \\ x_3 & -x_2 & 0 \\ \end{pmatrix}\]

Theorem. There is a \(\alpha_J\in \wedge^3 V\), such that

\[ J\subset \mathbb{P}^8 : \{ v\in \mathbb{P}^8 : \mathrm{rank}\Phi(\alpha_J)(v) \le 4\} \]

3-Descent

is thought as the following sequence

\[ J(\mathbb{Q})/3J(\mathbb{Q}) \to {\mathrm{Sel}}^{(3)}(J/\mathbb{Q}) \to \mathrm{Sha}(J/\mathbb{Q})[3] \]

\[{\mathrm{Sel}}^{(3)}(J/\mathbb{Q}) = \{ T\subset \mathbb{P}^8 : T\text{ twist of } J\subset \mathbb{P}^8, |3\cdot \theta|\}.\]